def test_partitions():
assert [npartitions(k) for k in range(13)] == [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77]
assert npartitions(100) == 190569292
assert npartitions(200) == 3972999029388
assert npartitions(1000) == 24061467864032622473692149727991
assert npartitions(2000) == 4720819175619413888601432406799959512200344166
assert npartitions(10000) % 10 ** 10 == 6916435144
assert npartitions(100000) % 10 ** 10 == 9421098519
def test_partitions():
assert [npartitions(k) for k in range(13)] == \
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77]
assert npartitions(100) == 190569292
assert npartitions(200) == 3972999029388
assert npartitions(1000) == 24061467864032622473692149727991
assert npartitions(2000) == 4720819175619413888601432406799959512200344166
assert npartitions(10000) % 10**10 == 6916435144
assert npartitions(100000) % 10**10 == 9421098519
def fast_but_wrong():
"""
http://mathworld.wolfram.com/PartitionFunctionPCongruences.html
and divisble by 10**6 = 2**6 * 5**6 (prime factorisation)
Watson (1938) then proved the general congruence
P(n) congru 0 (mod 5^a) if 24*n congru 1 (mod 5^a)
cad
24*n = b*5**6 + 1 = b*(15625) + 1
so we can deduce n
n = k*5**6 - 651 for k >= 1
and we want P(n) = b'*1 000 000
so P(k*5**6 - 651) = b'*1 000 000
"""
plist = [1,1,2,3,5]
def euler_p(n):
if n >= len(plist):
pn = 0
for k in range(1,n+1):
k1 = int(n - 1./2*k*(3*k-1))
k2 = int(n - 1./2*k*(3*k+1))
if k1 >= 0:
pn += ((-1)**(k+1))*(euler_p(k1))
if k2 >= 0:
pn += ((-1)**(k+1))*(euler_p(k2))
plist.append(pn)
return plist[n]
from sympy.ntheory import npartitions
k = 1
while True:
n = k*5**6 - 651
#n = k
# # p(7*k + 5) congru 0 (mod 7)
# if (n-5)%7==0:
# print "p(%d) divisible par 7" % n
# # p(11*k + 6) congru 0 (mod 11)
# elif (n-6)%11==0:
# print "p(%d) divisible par 11" % n
# # p(11**3 * 13 * k + 237) congru 0 (mod 13)
# elif (n-237)%(13*11**3)==0:
# print "p(%d) divisible par 13" % n
# else:
p = npartitions(n)
# show that there is often a error of 1 in the npartitions result
# if p != euler_p(n):
# print "NANNNN", p, euler_p(n)
# raw_input()
if (p-1)%N==0 or p%N==0 or (p+1)%N==0:
print "Potential", n, p
print " compute euler value"
print "", euler_p(n)
raw_input("continue ?")
else:
print "OUT", n, str(p)[-10:]
k += 1
return n,p
from sympy.ntheory import npartitions print(npartitions(100) - 1)
from sympy.ntheory import npartitions
import sys
for n in range(0, sys.maxsize**10):
np = npartitions(n)
if np % 1000000 == 0:
print(n)
print(np)
if n % 10000 == 0:
print(n)
print(np)
# Let p(n) represent the number of different ways in which n coins can be separated into piles.
# For example, five coins can be separated into piles in exactly seven different ways, so p(5)=7.
# OOOOO
# OOOO O
# OOO OO
# OOO O O
# OO OO O
# OO O O O
# O O O O O
# Find the least value of n for which p(n) is divisible by one million.
from sympy.ntheory import npartitions
for i in range(1, 100000):
x = npartitions(i)
if x % 1000000 == 0:
print('resultaat: ', x)
print('n: ', i)
break
if i % 1000 == 0:
print('tussenstand: ', i, x)